Factoring Polynomial Questions

Question

How do you decide whether to use synthetic division or the factor theorem to help you factor a polynomial?

Please help me answer.

Answer 1

Synthetic division is helpful in factoring only if you have a factor or guess at a factor, in which case it will tell you if the supposed factor is indeed a factor, and reduce the problem to that of a lower degree polynomial.

The factor theorem can help you find a factor, if you notice that some particular value must be a zero of the polynomial. For example, in $$ P(x) = x^3 + 53x^2 - 26x -28 $$ since the coefficients add to zero, you know that $x=1$ is a root, so $x-1$ must be a factor. Then synthetic division can tell you that the polynomial is $$ P(x) = (x-1) (x^2 + 54x + 28) $$ Similarly, if the coefficients using alternating addition and subtraction give zero, then the polynomial is divisible by $(1+x)$. Other such techniques exist as well.

The third tool in your box for factoring is the knowledge that any integer root must divide the constant term in the polynomial; elementary problems often are constructed to have nice integer roots.

The fourth tool is that a quadratic polynomial can be solved using the quadratic formula. (Yes, there are formulas for cubic and quartic polynomials, but they are not practical to use. For example, if you multiply $(x-2)(x+7)(x-8)$ and try using the cubic formula, it takes a staggering amount of work to see the cancellations that leave you with $2$, $-7$ and $8$ as solutions.)